Suppose I have two states $\rho_{AB}$ and $\sigma_{AB}$ such that the marginals $\rho_A = \sigma_A$. What is the most general operation that could have acted on $\rho$ to output $\sigma$?

For example, if there was no reduced state constraint, I think one can always find a unitary $U_{ABC}$ where $C$ is a purifying system such that $U_{ABC}\vert\rho_{ABC}\rangle = \vert\sigma_{ABC}\rangle$.

Now enforcing $\rho_A=\sigma_A$, is it true that there exists some unitary $U_{BC}$ that achieves $(I_A\otimes U_{BC})\vert\rho_{ABC}\rangle = \vert\sigma_{ABC}\rangle$? Or some other way to achieve such a transformation?